Average Error: 58.0 → 0.6
Time: 22.7s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{(x \cdot \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right) + \left(\frac{1}{60} \cdot {x}^{5}\right))_*}{2}\]

Error

Bits error versus x

Derivation

  1. Initial program 58.0

    \[\frac{e^{x} - e^{-x}}{2}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]

  3. Simplified0.6

    \[\leadsto \frac{\color{blue}{(x \cdot \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right) + \left({x}^{5} \cdot \frac{1}{60}\right))_*}}{2}\]

  4. Final simplification0.6

    \[\leadsto \frac{(x \cdot \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right) + \left(\frac{1}{60} \cdot {x}^{5}\right))_*}{2}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 13.1s)Debug log

start245.0ms

Algorithm
intervals

setup49.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.1b

localize12.0ms

Local error

Found 1 expressions with local error:

5.7b
(- (exp x) (exp (- x)))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
1 → 28
Calls

1 calls. Slowest were:

5.0ms
(- (exp x) (exp (- x)))

series77.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

77.0ms
(- (exp x) (exp (- x)))

simplify663.0ms

Counts
27 → 31
Calls

27 calls. Slowest were:

69.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
55.0ms
(fma 1 (exp x) (- (* (exp (- x)) 1)))
39.0ms
(fma (sqrt (exp x)) (sqrt (exp x)) (- (* (sqrt (exp (- x))) (sqrt (exp (- x))))))

prune292.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize28.0ms

Local error

Found 2 expressions with local error:

0.0b
(* (pow x 5) 1/60)
0.0b
(fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 23
Calls

2 calls. Slowest were:

4.0ms
(* (pow x 5) 1/60)
0.0ms
(fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60))

series63.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

35.0ms
(* (pow x 5) 1/60)
28.0ms
(fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60))

simplify154.0ms

Counts
8 → 29
Calls

8 calls. Slowest were:

49.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
48.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
39.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))

prune389.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize27.0ms

Local error

Found 4 expressions with local error:

0.7b
(cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)))
0.7b
(cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)))
0.7b
(cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)))
0.5b
(* (cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60))) (cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60))))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

10.0ms
(* (cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60))) (cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60))))
1.0ms
(cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)))
1.0ms
(cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)))

series646.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

168.0ms
(cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)))
165.0ms
(cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)))
159.0ms
(* (cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60))) (cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60))))
153.0ms
(cbrt (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)))

simplify4.9s

Counts
41 → 81
Calls

41 calls. Slowest were:

502.0ms
(* (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)) (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)))
496.0ms
(* (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)) (fma x (fma 1/3 (* x x) 2) (* (pow x 5) 1/60)))
446.0ms
(+ (exp (* 1/3 (- (log 1/3600) (* 10 (log (/ 1 x)))))) (+ (* 40/3 (/ (exp (* 1/3 (- (log 1/3600) (* 10 (log (/ 1 x)))))) (pow x 2))) (* 320/9 (/ (exp (* 1/3 (- (log 1/3600) (* 10 (log (/ 1 x)))))) (pow x 4)))))

prune1.3s

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 4 expressions with local error:

5.7b
(- (exp x) (exp (- x)))
5.7b
(- (exp x) (exp (- x)))
5.7b
(- (exp x) (exp (- x)))
0.1b
(cbrt (- (exp x) (exp (- x))))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

10.0ms
(- (exp x) (exp (- x)))
10.0ms
(- (exp x) (exp (- x)))
10.0ms
(- (exp x) (exp (- x)))

series305.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

180.0ms
(cbrt (- (exp x) (exp (- x))))
49.0ms
(- (exp x) (exp (- x)))
39.0ms
(- (exp x) (exp (- x)))
38.0ms
(- (exp x) (exp (- x)))

simplify2.5s

Counts
91 → 113
Calls

91 calls. Slowest were:

175.0ms
(- (+ (exp (* 1/3 (+ (log 2) (log x)))) (* 1/18 (* (pow x 2) (exp (* 1/3 (+ (log 2) (log x))))))) (* 1/3240 (* (pow x 4) (exp (* 1/3 (+ (log 2) (log x)))))))
170.0ms
(cbrt (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))))
123.0ms
(cbrt (- (pow (exp x) 3) (pow (exp (- x)) 3)))

prune1.2s

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.0b

regimes185.0ms

Accuracy

0% (0.6b remaining)

Error of 0.6b against oracle of 0.0b and baseline of 0.6b

bsearch2.0ms